Methods and apparatus for improved error and erasure correction in a Reed-Solomon date channel

ABSTRACT

Methods and associated structures for improved erasure correction and detection in digital communication channels utilizing modified Reed-Solomon decoding of encoded digital data. Methods and associated apparatus in accordance with features and aspects hereof perform Galois Field element generation in descending order for Reed-Solomon erasure detection and correction. Real time computation of Galois Field elements in descending order as required for erasure detection and correction features and aspects hereof eliminates the need for costly, complex, large, high speed lookup tables as previously practiced in the art for storing Galois Field element values pre-computed in the same ascending order of reception of the encoded code words. Features and aspects hereof may thus be applied in digital read channel applications including, for example, digital telecommunications receive/read channels and digital data storage read channels.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to error processing in a digital datachannel and more specifically relates to improved methods and structuresfor real time computation of an erasure polynomial to detect and correctdata errors from erasure conditions in a digital data channel usingReed-Solomon encoding/decoding.

2. Discussion of Related Art

In digital communication systems it is generally known to transmit orstore digital data encoded in forms that improve reliability in readingstored data or receiving transmitted data. Noise on the digitalcommunication channel or other symbol interference generated within thechannel or induced in the channel by external signals may causeun-encoded information to be frequently erroneously transmitted or read.Transforming the digital data into encoded sequences based onmathematical spreading of the encoded symbols and generatingcorresponding error correction and detection codes (“ECC”), such adigital channel may be used in a manner far more immune to noise andinterference.

As used herein, “communication” channel includes any receivingcomponents of a digitally encoded channel. For example, a receiver in adigital communication system or a read head in a data storage system areboth examples of communication channels that may encounter problemsdiscussed herein and may beneficially apply the features and aspectshereof. Thus “communication channel” and “read channel” may beunderstood as synonymous in the context of this application.

Exemplary of such digital channels are digital communication channelssuch as computer or telecommunications network communication media aswell as data storage systems wherein data is transmitted to the storagedevice for storage on a recordable medium as encoded code words and readback from the recordable medium. In communication digital channels,noise and interference may be caused by the analog modulation of theencoded signals and associated analog noise in the ambient environmentin which the transmission is performed. In storage applications, thedigital channel modulates encoded digital information and associatederror correction information onto the associated recordable medium asmagnetic flux changes and/or optical property changes on the recordablemedia. Reading the previously recorded information back, the digitalread channel must cope with inter-symbol interference and other aspectsof the analog recording of the encoded digital information associatedwith the magnetic and/or optical storage medium.

It is generally known in the art to utilize Reed-Solomon encodingtechniques for high-speed digital communication channels andhigh-density digital data storage applications. In general, Reed-Solomonencoding techniques encode and embed error correction information alongwith each encoded unit of transmitted or stored information. Thus,Reed-Solomon encoding as applied in digital communication channels(e.g., telecommunications and/or data storage applications) provides acompact encoding of information including highly effective errorcorrection. Sequences of encoded code words are expected and/orprecluded in accordance with the encoding techniques of Reed-Solomon.Thus, as each code word is decoded errors may be detected from the errorcorrection information associated with each code word based on thelikelihood of particular code words being encountered in particularsequences.

In addition to correction of erroneous digital data received orretrieved by using Reed-Solomon encoding, other modifications to knownReed-Solomon techniques have been employed to further enhance detectingand correcting erasures of previously stored data. In general, modifiedReed-Solomon techniques may also detect and correct erasures of encodeddata as distinct from merely erroneous transmission or retrieval ofpreviously encoded information.

As well known to those of ordinary skill in the art, modifiedReed-Solomon techniques used to correct both errors and erasures arebased upon a discrete time based polynomial expression derived fromapplication of Galois Fields. As used for erasure detection andcorrection, the Galois Field generator for typical 10-bit code words maybe expressed as a discrete time polynomial as follows:m(x)=x ¹⁰ +x ³+1

Those of ordinary skill in the art will recognize many other discretetime polynomials that may be utilized for generation of Galois Fieldvalues. Thus the above polynomial is intended merely as exemplary of onesuch discrete time polynomial useful for Galois Field value generation.

A typical Galois Field generator circuit may be implemented as a shiftregister structure coupled with a summing component as generally knownin the field. Exemplary of a known shift register structure inaccordance with the above discrete time equation, the Galois Fieldgenerator GF(2¹⁰) may be implemented as a left shift register andsumming junction well known to those of ordinary skill in the art. Sucha typical Galois Field generator computes the Galois Field (“GF”)element values on each clock cycle in ascending order since eachsubsequent GF element depends on the previous element. However, asapplied to erasure correction as distinct from error correction usingReed-Solomon encoding, the Galois Field elements are utilized in reverseorder—e.g., in descending order relative to the ascending order of thereceived/retrieved code words.

As presently practiced it is impractical to compute each Galois Fieldelement for erasure detection and correction in real time as it isrequired in the Reed-Solomon decoding process since utilization of theelements in descending order requires re-computation of each earlierelement. Such duplicative re-computation of Galois Field elementsrenders the computational process too slow for real time generation ofthe Galois Field elements in high speed communication application orhigh density storage applications. Thus, as presently practiced, theGalois Field elements are generated in ascending order and saved in alookup table comprised of high speed memory (e.g., a register file orhigh speed memory circuits) to assure rapid access to the pre-computedGF element values.

It is a problem in such applications that the lookup table used to storepre-computed Galois Field elements may be large. Since the utilizedmemory structures must be high speed (e.g., a register array) the lookuptable circuits may consume substantial circuitry and associated powerwithin the application circuit design for the digital communicationchannel. For example, a typical GF generator such as GF(2¹⁰) may requirestorage of up to 1023 10-bit GF element values. As recording density ortransmission speeds increase in the digital communication channel, evenlarger lookup table structures may be required. Such substantial storagerequirements for high speed memory components such as register filesrepresent significant complexity and cost in design of the digitalcommunication channel circuits.

It is evident from the above discussion that a need exists for animproved communication channel design useful in Reed-Solomon erasuredetection and correction applications to permit real time error anderasure detection and correction while minimizing circuit complexity andcost area.

SUMMARY OF THE INVENTION

The present invention solves the above and other problems, therebyadvancing the state of the useful arts, by providing apparatus andmethods for processing error detection and correction information todetect and correct both errors and erasures in real-time as Reed-Solomonencoded data is received in a digital communication channel. Featuresand aspects hereof generate Galois Field elements in descending orderfor use in detecting and correcting erasures in the received,Reed-Solomon encoded information. The real time descending Galois Fieldgeneration obviates the need for lookup tables and associated complex,costly, high speed memory circuits in a Reed-Solomon decoder aspresently practiced in the art.

In one aspect, a method is provided for processing received data in adigital communication channel. The method includes receiving a sequenceof code words representing encoded digital data. The method thenprovides for decoding the encoded digital data to generate decodeddigital data to be utilized in further data processing. The step ofdecoding further includes correcting erasures in the sequence of codewords by applying a discrete time polynomial thereto such that thediscrete time polynomial utilizes Galois Field elements determined fromthe sequence code words, and generating the Galois Field elements inreal time substantially concurrently with reception of the sequence ofcode words such that the Galois Field elements are generated indescending order relative to the order of reception of the sequence ofcode words.

In another aspect, a communication channel is provided including areceiver for receiving a sequence of words representing encoded digitaldata. The channel also includes a decoder coupled to receive a sequenceof code words representing encoded digital data. The decoder is adaptedto generate corresponding decoded digital data and further includes anerasure correction element adapted to detect and correct erasures in thereceived sequence of code words by applying a discrete time polynomialto the sequence of code words such that the discrete time polynomialutilizes Galois Field elements determined from a received code word. Thedecoder also includes a Galois Field generator adapted to generate theGalois Fields elements in real time substantially concurrently withreception of the sequence of code words such that the Galois Fieldelements are generated in descending order relative to the order ofreception of the sequence of code words.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram showing an exemplary system embodying featuresand aspects hereof to improve erasure detection and correction inReed-Solomon erasure detection and correction.

FIG. 2 is a flowchart describing exemplary operation of a system such asthat of FIG. 1 in accordance with features and aspects hereof to improveReed-Solomon erasure detection and correction.

FIG. 3 depicts an exemplary communication channel operable in accordancewith features and aspects hereof to provide enhanced erasure detectionand correction for Reed-Solomon encoded information.

FIG. 4 depicts an exemplary right shift register adapted in accordancewith features and aspects hereof to generate Galois Field values indescending order.

FIG. 5 depicts and exemplary structure for recursive computation of anerasure locator polynomial in accordance with features and aspectshereof.

FIG. 6 is a block diagram providing an exemplary system for enhancederasure detection and correction in accordance with features and aspectshereof.

DETAILED DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram showing a system 100 embodying features andaspects hereof System 100 may be any device or system in which digital,encoded data is received or retrieved to be decoded and utilized. Forexample, system 100 may be a receiving component adapted to receiveencoded digital data from a transmitting device of a digitalcommunication system. The transmission medium (not shown) may beelectrical impulses carried over wires, radio frequency transmissions,optical modulations, etc. Or, for example, system 100 may be a componentof a storage device such as a rotating magnetic or optical disk drive inwhich encoded digital data is stored on a recordable medium modulated aschanges in the magnetic or optical properties of the recordable medium.System 100 then retrieves the previously stored, encoded, digital data.As is well known in the art, encoding of such digital data helps toreduce error rates in the received or retrieved data. In general, suchdigital encoding techniques encode and spread units of data to betransmitted or stored over a larger range of encoded code words in sucha manner that a decoding system 100 may detect and correct errors aswell as erasures in the received or retrieved data based on expectedsequences of code words and disallowed sequences of code word values.

In general, system 100 may include a receiver or transducer element 102for receiving the transmitted, encoded digital data or for sensing themodulation changes of the previously recorded, stored data. The receivedor retrieved sequence of bits representing encoded digital data aregrouped into symbols or code words of fixed or variable length dependingon the particular encoding techniques utilized. Communication channel104 thus receives a sequence of code words representing the encodeddigital data. Decoder 108 within communication channel 104 decodes thesequence of code words received by communication channel 104 and appliesthe decoded digital data to controller 106 for further utilization ofthe decoded information.

In accordance with features and aspects hereof, decoder 108 mayimplement Reed-Solomon techniques for decoding the encoded sequence ofcode words. Reed-Solomon encoding/decoding techniques are well known tothose of ordinary skill in the art and are commonly applied in a varietyof digital communication systems and data storage systems. Reed-Solomonencoding/decoding techniques provide for highly effective errordetection and correction and for erasure detection and correction.Decoder 108 may therefore preferably be a Reed-Solomon decoderimplementing Reed-Solomon decoding techniques including associated errorcorrection and erasure correction. Decoder 108 therefore may includeerror detection and correction element 110 and erasure detection andcorrection element 112.

Error detection and correction element 110 is operable in accordancewith well-known Reed-Solomon decoding techniques to allow for correctionof a variety of errors detected in the received sequence of encoded codewords. In general, such error detection techniques implement computationof a polynomial function of the received code words and generated GaloisField values. Galois Field values are generally utilized in theReed-Solomon encoding/decoding as a basis for the spreading of thedigital data into encoded code words and thus for detecting off avariety of errors in the received code words. For detection andcorrection of errors by element 110, the well-known polynomial for sucherror correction utilizes Galois Fields computed/generated in an orderthat essentially matches the order in which encoded code words arereceived. In other words, a first Galois Field element may be computedin association with the first received code word, the second GF value iscomputed with the second received code word, etc. Error detection andcorrection element 110 then uses the GF values computed in the sameorder as the received code words for correcting various bit errors inthe received code words.

As noted above and as well known in the art, erasure detection andcorrection in association with Reed-Solomon encoded information requiresutilization of a polynomial that uses Galois Field values thatcorrespond to the reverse order of received encoded code words. Ingeneral, erasure detection and correction of Reed-Solomon encodedinformation presumes that the sequence of encoded code words islogically organized into fixed size blocks. The polynomial used for sucherasure detection and correction therefore presumes knowledge of thispredetermined block size in generating and using the GF values. In thecontext of storage device applications of Reed-Solomon encoding anddecoding, the fixed block size may correspond to a block or sector sizeassociated with the recordable medium of the storage device. Thepolynomial used for detecting and correcting erasure in a first receivedcode word of the sequence of code words requires a Galois Field valuecomputed by the operation of the Galois Field generator based onreception of the last code word of the sequence of code words of thefixed size block, the erasure detection and correction for the secondcode word uses the GF value associated with the next to last code wordof the block. In other words for a block of N code words, as code words1 . . . N are received, the GF value required for erasure detection andcorrection for each received code word is that which corresponds to codewords N . . . 1.

One known method of erasure detection and correction in Reed-Solomonencoding is to operate a Galois Field generator circuit in such a manneras to generate all possible GF values useful for the block of encodedcode words. The pre-computed GF values are stored in a table structureor register file for subsequent access by the erasure locator polynomialcomputation structures. Thus, in a sequence of code words representing ablock of N units of data, in response to receipt of the first code wordof a block, the erasure detection and correction polynomial computationstructure may access the GF values from the table or register filecorresponding to the N^(th) entry. More generally, such known tablestructures or register files may access a pre-computed GF values attable entry index N−i (where i=1 . . . N) the index of the received codeword in the sequence of code words.

As noted above currently practiced techniques utilizing a tablestructure or register file for storing pre-computed Galois Field valuesis costly and complex. The size of such a table may represent asignificant cost and complexity in computational circuits used for theReed-Solomon decoding components of system 100. For example, a commonapplication of Reed-Solomon encoding/decoding techniques generates a10-bit code word to represent each 8-bit byte of un-encoded digitaldata. Such a ten bit code word requires 2¹⁰ GF values for polynomialcomputation for erasure detection and correction. Thus, a table orregister file structure used to store pre-computed Galois Field valueswould require 1024 entries of 10 bits each. Storage device applicationsof Reed-Solomon encoding/decoding techniques are starting to use largerand larger block sizes. In addition, the size of the code word used forencoding bytes of data is also increasing to further spread the encodingof the data and thus permit better error correction. Thus a tablestructure or register file would require still more storage space forretaining the pre-computed GF values for larger blocks and for largercode words. In addition, performance of such a table structure orregister file is critical to performance of the Reed-Solomon decoder108. The table structure or register file cannot utilize slower speed,lower cost, memory structures or the required performance for high speedreception or retrieval cannot be attained. Thus, the table structure orregister file must utilize high speed, higher cost memory devices and/orregister file structures.

Erasure detection and correction element 112 of FIG. 1 is enhanced inaccordance with features and aspects hereof to provide real timecomputation/generation of required Galois Field values for Reed-Solomonerasure detection and correction without the need for such high speed,high cost memory devices and/or register file structures for storingpre-computed GF values. In accordance with features and aspects hereof,a Galois Field generator generates GF values as required for erasuredetection and correction in real time substantially concurrent withreception of the encoded code words and associated decoding thereof. Bycomparison to other known GF generators as used for error detection andcorrection rather than erasure detection and correction, the GaloisField generator in accordance with features and aspects hereof uses aright shift register structure to compute each GF value in reverse orderof the received code words. In other words, as the first code word of asequence of code words in a block of size N is received, a Galois Fieldcorresponding to the N^(th) GF value is computed in real timesubstantially concurrently with reception/retrieval and decoding of thefirst code word. Details of an exemplary Galois Field generator inaccordance with features and aspects hereof are presented further hereinbelow.

As noted above, system 100 of FIG. 1 may be useful in the context ofdigital communication systems where a transmitting device (not shown)transmits an encoded sequence of code words (logically organized intofixed size blocks) for reception and decoding by system 100. Stillfurther, as noted above, system 100 of FIG. 1 may be usefully applied inthe context of a data storage device (e.g., magnetic and/or optical diskdrives or other storage devices). Writing/recording features of such astorage device (not shown in FIG. 1) may encode users supplied data inaccordance with Reed-Solomon encoding techniques and record the encodeddata by modulating magnetic and/or optical properties of a recordingmedium of the storage device. Thus the system 100 including the enhancederasure detection and correction element 112 may be advantageouslyapplied to a wide variety of high speed digital data communicationssystems and high density digital data recording systems.

Those of ordinary skill in the art will readily recognize a wide varietyof additional components within a fully operational system 100. Inparticular, a variety of digital and analog devices may be present in areceiver/transducer element 102. A variety of digital processingelements may also be required within an operational communicationchannel 104 to perform all other functionality of the communicationchannel in addition to the decoding function of element 108. Those ofordinary skill in the arts will readily recognize such additionalelements as may be required for a fully operational system 100 and willfurther recognize that these additional components are eliminated fromFIG. 1 merely for simplicity and brevity of this description.

FIG. 6 is a block diagram of another apparatus in accordance withfeatures and aspects hereof showing the key functions of a system 600operable in accordance with features and aspects hereof. Receivedmessage code words are received in syndrome computation element 602.Erasure locator indicators (e.g., from a signal of the communicationchannel) are receive at erasure polynomial calculation element 604.Error detection and correction element 606 then receives decodedsyndromes from element 602 and erasure polynomial values correspondingthereto from element 604 and applies Reed-Solomon decoding and errorcorrection to the received syndromes. Erasure polynomial calculationelement 604 is enhanced in accordance with features and aspects hereofto generate requisite Galois Field values in descending order asdiscussed above to obviate the need of prior solutions for large, highperformance lookup table structures to retrieve pre-determined GaloisField values.

FIG. 2 is a flowchart broadly describing operation of a system such assystem 100 of FIG. 1 in accordance with features and aspects hereof.Element 200 represents processing associated with the receipt of eachcode word. As noted, code words are received in a first order orsequence and logically organized in fixed size blocks of N code words.The received code words may be encoded, for example, in accordance withknown Reed-Solomon encoding techniques including a variety of modifiedversions of Reed-Solomon encoding that provide for enhanced errordetection and correction as well as enhanced erasure detection andcorrection.

For each received code word, elements 202 through 209 are next operableuntil element 209 determines that the entire block of N code words hasbeen received. Specifically, element 202 is operable to receive the nextencoded code word. Element 204 is operable to detect whether a flagassociated with the received code word indicates that the received codeword is possibly invalid due to an erasure condition. If the receivedcode word appears valid (not erased), element 209 then determines ifthis was the last code word expected in the block and loops back toelement 202 to await receipt of the next code word. If element 204determines that the next received code word is likely invalid due to apossible erasure, element 206 is next operable to generate acorresponding Galois Field value for use in computation of the erasurelocating and correction polynomial. As noted above, features and aspectshereof permit real time computation of the Galois Field value requiredfor this received code word by computing the GF values in the oppositeor descending order as compared to the order in which code words arereceived. Thus, features and aspects hereof permit real time computationof the GF value without the need for high speed, high cost memorystructures or register file structures for storing a large number ofpre-computed GF values.

Utilizing the Galois Field value computed by element 206, element 208 isnext operable to compute the erasure polynomial for the particularerasure in the newly received code word. After receiving the block ofcode words, element 209 is again operable to determine if more codewords are expected in the block of code words. If so, processing loopsback to element 202 to await a next code word.

When all code words of the block have been received and the erasurelocator polynomials have been computed for each possibly erased codeword (using the generated GF values), element 210 is then operable toperform standard error detection and correction of the received block ofcode words. Element 212 is lastly operable to utilize the decoded,corrected data value as appropriate for the particular systemapplication. Those of ordinary skill in the art will readily recognizeother detailed processing steps and elements useful within. NormalReed-Solomon decoding techniques and normal Reed-Solomon error detectingand correcting techniques are applied within element 210 after thesyndromes and erasure polynomial computation of the block of receivedcode words. Such Reed-Solomon processing and error detection andcorrection techniques and structures are well known to those of ordinaryskill in the art and well documented in the prior art.

FIG. 3 depicts an exemplary communication channel 300 operable inaccordance with features and aspects hereof to provide enhanced at theerasure detection and correction. Communication channel 300 receives asequence of encoded code words r_(i), i=1 . . . N where N is the size ofthe logical block of code words. Received code words are decoded andcorrected in accordance with Reed-Solomon encoding/decoding techniques.Thus, communication channel 300 includes, among other elements,Reed-Solomon decoder element 302 operable to decode each received codeword. Techniques and structures for normal decoding of each receivedcode word are well known to those of ordinary skill in the art. Suchstandard techniques include correction of errors other than erasureerrors as well as prior known techniques for correcting erasuresdetected in received code words. As noted above, prior erasure detectionand correction techniques required high speed, high cost memory devicesand/or register file circuit structures to store a potentially largenumber of pre-computed Galois Field values for use in erasure detectionand correction. In accordance with features and aspects hereof erasuredetection and correction element 304 is co-operable with descendingorder Galois Field generator 306. Erasure detection and correctionelement 304 utilizes a well known polynomial for computing for detectinga likely erasure in a received code word and for correcting same whendetected. The following polynomial is well known in Reed-Solomondecoding as useful for an erasure locator polynomial.

${\Gamma(x)} = {\prod\limits_{j = 0}^{\rho}\left( {1 - {x\;\alpha^{j_{n}}}} \right)}$

where: ρ is the number of erasures inside the received n code words, andα^(j) ^(n) is the Galois Field element associated with the erased codeword j inside the receiving n code words. The above well-knownpolynomial expression can also be expressed in a recursive computationas follows:Γ^((i))(x)=Γ^((i−1))(x)(1−α^(n−i) x)which may be simplified to:Γ_(j) ^((i))=Γ_(j) ^((i−1))−Γ_(j−1) ^((i−1))α^(n−i)

where (i) is the cycle time. For example, Γ_(j) ^((i)) represents thevalue of register j in FIG. 5 noted below at (i) moment; Γ_(j) ^((i−1))represents the value of register j at time (i−1), and α^(n−i) is theGalois Field GF(2^(m)) element associated with the received symbolr_(i). Where i=1 . . . n.

FIG. 5 is a diagram of the discrete time computation logic associatedwith the recursive expression of the erasure locator polynomial.Erasure_flg_en is a signal generated external to the Reed-Solomondecoder (e.g., from the receiver/transducer or other elements of thecommunication channel device circuit) indicating that the current codeword “i” in the sequence of code words i=1 . . . n is likely invalid dueto an erasure. The upper portion of FIG. 5 graphically depicts thesimplified recursive equation above and can be described as: register jat time i equals its previous value at time i−1 minus the value inregister j−1 at time i−1 multiplied by the Galois Field element α^(n−i).The lower portion of FIG. 5 calculates Γ(x). For example, when there isno erasure in n code words, only Γ₀ is “1”, all other Γ registers have“0” values. When one erasure is sensed, Γ₀ is still “1”, Γ₁ willcalculated by the above recursive equations. In the case of twoerasures, Γ₀ is “1”, Γ₁ and Γ₂ are calculated as in the above recursiveequations, and so on. FIG. 5 therefore represents the erasure polynomialregister values when ρ erasures are sensed in n code words.

As can be seen in the above recursive equations and FIG. 5, computationof the erasure locator polynomial requires a Galois Field valuegenerated in the reverse or descending order relative to the order inwhich symbols are received (e.g., generating GF values for indices n . .. 1 as code words are received in the sequence 1 . . . n). Thedescending order of generation of Galois Field values by element 306 maybe implemented by an m-stage right shift register with predefined tapsfor the field generator m(x). For example in the case of a 10-bitReed-Solomon code word, m(x)=x¹⁰+x³+1 expresses the operation of thedescending Galois Field generator GF(2¹⁰). FIG. 4 graphically depictssuch an m-stage right shift register 400 with a predefined tap 402.Initialization of such a descending GF(2¹⁰) field generator may beperformed as: α⁽⁰⁾=α^(n−1), where n is the receiving symbol degree.

Logic circuits to implement the components shown by FIGS. 4 and 5 willbe readily apparent to those of ordinary skill in the art through use ofsimple register and shift register structures, summing junctions, andproduct junctions. Standard discrete, combinatorial logic elements maybe used to implement such a structure as well as custom designedintegrated circuit structures. A variety of equivalent circuitstructures will be readily apparent to those of ordinary skill in theart further. Further, ubiquitous bit clock signals (not shown in FIGS. 4and 5) are applied in accordance with the particular application datarate as transmitted code words are received and/or sensed by anappropriate transducer. Clocking signals associated with logic of FIGS.4 and 5 will be readily apparent to those of ordinary skill in the art.

While the invention has been illustrated and described in the drawingsand foregoing description, such illustration and description is to beconsidered as exemplary and not restrictive in character. One embodimentof the invention and minor variants thereof have been shown anddescribed. Protection is desired for all changes and modifications thatcome within the spirit of the invention. Those skilled in the art willappreciate variations of the above-described embodiments that fallwithin the scope of the invention. In particular, those of ordinaryskill in the art will readily recognize that features and aspects hereofmay be implemented equivalently in electronic circuits or as suitablyprogrammed instructions of a general or special purpose processor. Suchequivalency of circuit and programming designs is well known to thoseskilled in the art as a matter of design choice. As a result, theinvention is not limited to the specific examples and illustrationsdiscussed above, but only by the following claims and their equivalents.

1. In a digital communication system comprising a read channel, a methodof processing received data in a digital communication channel, themethod comprising: receiving a sequence of code words representingencoded digital data; decoding the encoded digital data to generatedecoded digital data to be utilized in further data processing, whereinthe step of decoding further comprises: correcting erasures in thesequence of code words by applying a discrete time polynomial theretowherein the discrete time polynomial utilizes Galois Field elementsdetermined from the sequence code words; and generating the Galois Fieldelements in real time substantially concurrently with reception of thesequence of code words wherein the Galois Field elements are generatedin descending order relative to the order of reception of the sequenceof code words, wherein the steps of receiving, decoding, correcting, andgenerating are all performed by the digital communication channel. 2.The method of claim 1 wherein the step of decoding further comprisesapplying the sequence of code words to a Reed-Solomon decoder.
 3. Themethod of claim 2 wherein the step of correcting further comprisesapplying the sequence of code words to an erasure locator polynomial inaccordance with the Reed-Solomon decoder.
 4. The method of claim 3wherein the erasure locator polynomial is:${\Gamma(x)} = {\prod\limits_{j = 0}^{\rho}\left( {1 - {x\;\alpha^{j_{n}}}} \right)}$where: α^(j) ^(n) is the Galois Field element associated with the erasedcode word j inside the receiving n code words, ρ is the number oferasure flags received.
 5. The method of claim 1 wherein the step ofgenerating the Galois Field elements further comprises operating aGalois Field generator circuit comprising a right shift register with afeedback connection operable in accordance with a pre-defined tap. 6.The method of claim 5 wherein each code words is 10 bits and wherein theright shift register is a 10-bit register operable in accordance withthe pre-defined tap defined by the discrete time polynomialm(x)=x¹⁰+x³+1 where x is a 10 bit code word.
 7. The method of claim 1wherein the step of receiving further comprises receiving the sequenceof code words from a read head transducer of a storage device.
 8. Themethod of claim 1 wherein the step of receiving further comprisesreceiving the sequence of code words from a receiver of a communicationdevice.
 9. A read channel comprising: a receiver for receiving asequence of words representing encoded digital data; and a decodercoupled to receive a sequence of code words representing encoded digitaldata, the decoder adapted to generate corresponding decoded digital datawherein the decoder further comprises: an erasure correction elementadapted to detect and correct erasures in the received sequence of codewords by applying a discrete time polynomial to the sequence of codewords wherein the discrete time polynomial utilizes Galois Fieldelements determined from a received code word; and a Galois Fieldgenerator adapted to generate the Galois Fields elements in real timesubstantially concurrently with reception of the sequence of code wordswherein the Galois Field elements are generated in descending orderrelative to the order of reception of the sequence of code words. 10.The read channel of claim 9 wherein the decoder is a Reed-Solomondecoder.
 11. The read channel of claim 10 wherein the erasure correctionelement is operable in accordance with the erasure locator polynomial:${\Gamma(x)} = {\prod\limits_{j = 0}^{\rho}\left( {1 - {x\;\alpha^{j_{n}}}} \right)}$where: α^(j) ^(n) is the Galois Field element associated with the erasedcode word j inside the receiving n code word, ρ is the number of erasureflags received.
 12. The read channel of claim 9 wherein the Galois Fieldgenerator further comprises a right shift register with a feedbackconnection operable in accordance with a pre-defined tap.
 13. The readchannel of claim 12 wherein each code words is 10 bits and wherein theright shift register is a 10-bit register operable in accordance withthe pre-defined tap defined by the discrete time polynomialm(x)=x¹⁰+x³+1 where x is a 10 bit code word.
 14. The read channel ofclaim 9 wherein the read channel is adapted to receive the sequence ofcode words from a read head transducer of a storage device.
 15. The readchannel of claim 9 wherein the read channel is adapted to receive thesequence of code words from a receiver of a communication device.
 16. Adata storage device comprising: a recordable medium; a read headtransducer associated with the recordable medium for reading encodeddigital data as a sequence of code words previously recorded on therecordable medium; a read channel coupled to the read head transducer toreceive the sequence of code words and adapted to decode the sequence ofcode words to generate decoded data; and a controller coupled to theread channel and adapted to receive the decoded data and further adaptedto process the received decoded data, wherein the read channel furthercomprises: a Reed-Solomon decoder for decoding the sequence of codewords, wherein the Reed-Solomon decoder further comprises: an erasurecorrection circuit for correcting errors detected in the sequence ofcode words as caused by erasure of one or more code words, wherein theerasure correction circuit further comprises: a Galois Field generatorfor generating a sequence of Galois Field elements each elementcorresponding with a code word of the sequence of code words, whereinthe Galois Field generator is operable substantially concurrently withthe reception of the sequence of code words, and wherein the GaloisField generator is further adapted to generate the Galois Field elementsin descending order relative to the ascending order of the receivedsequence of code words.
 17. The data storage device of claim 16 whereinthe erasure correction circuit is operable in accordance with theerasure locator polynomial:${\Gamma(x)} = {\prod\limits_{j = 0}^{\rho}\left( {1 - {x\;\alpha^{j_{n}}}} \right)}$where: α^(j) ^(n) is the Galois Field element associated with the erasedcode word j inside the receiving n code words, ρ is the number oferasure flags received.
 18. The data storage device of claim 16 whereinthe Galois Field generator further comprises a right shift register witha feedback connection operable in accordance with a pre-defined tap. 19.The data storage device of claim 18 wherein each code words is 10 bitsand wherein the right shift register is a 10-bit register operable inaccordance with the pre-defined tap defined by the discrete timepolynomial m(x)=x¹⁰+x³+1 where x is a 10 bit code word.